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Значит:\n\n$$a < 0, \\qquad b > 0, \\qquad |a| > |b|.$$\n\nПроверим варианты.\n\n1. $ab^2 < 0$. Так как $b \\ne 0$, то $b^2 > 0$; произведение отрицательного $a$ на положительное $b^2$ отрицательно. Неравенство **верно**.\n2. $b - a < 0$. Здесь $b > 0$ и $-a > 0$, поэтому $b - a > 0$. Неверно.\n3. $ab > 0$. Произведение отрицательного и положительного числа отрицательно. Неверно.\n4. $a + b > 0$. Так как $|a| > |b|$, сумма имеет знак числа $a$, то есть $a + b < 0$. Неверно.\n\n**Ответ:** 1.","explanation":null,"score":1,"part":"1","metadata":{"answer_confidence":"high","answer_input_type":"choice_one","classify_confidence":"high","fipi_guid":"0D1B10CBCFE589FD463798CEC109AED2","fipi_short":"6F944F","qkind":"ILI_STD_SELECTONE","source":"fipi_oge_bank","tags":["координатная прямая","сравнение чисел","знаки чисел","числовые неравенства"]},"created_at":"2026-07-18T15:00:58.558218Z","updated_at":"2026-08-31T15:37:35.489242Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","name":"Неравенства","description":null},"tags":[{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33a6ace9-99be-4b5e-9b2a-7716d2d006e8","code":"tag-980","name":"сравнение чисел"},{"id":"536cfc98-1d12-4e75-a605-bf4e8216ce2e","code":"tag-780","name":"числовые неравенства"},{"id":"be65932e-76f1-4635-8f6a-d290fdfdf2f0","code":"tag-417","name":"координатная прямая"},{"id":"da06b24c-f6be-4d02-bbb8-4bfc6dd93f9e","code":"tag-382","name":"знаки чисел"}]}},{"problem_id":"2fc66a3a-7a78-4e3f-96b8-ce8e5b9d3f2c","collection_id":"82875418-ca33-4805-bb13-87a9b13be4ed","display_order":1,"metadata":{},"created_at":"2026-07-27T07:58:53.818369Z","updated_at":"2026-07-27T07:58:53.818369Z","problem":{"id":"2fc66a3a-7a78-4e3f-96b8-ce8e5b9d3f2c","short_id":20812,"task_number":7,"topic_id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","exam_id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","difficulty":2,"content_text":"На координатной прямой отмечены числа ![x](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/AD9FB8930F05A94C42BCD7D5100A4FC7/xs3qstsrcAD9FB8930F05A94C42BCD7D5100A4FC7_6_1734423903.png) и ![y](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/AD9FB8930F05A94C42BCD7D5100A4FC7/xs3qstsrcAD9FB8930F05A94C42BCD7D5100A4FC7_7_1734423903.png).\n\n![Координатная прямая: точка x слева от нуля и далеко от него, точка y справа от нуля и близко к нему](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/AD9FB8930F05A94C42BCD7D5100A4FC7/xs3qstsrcAD9FB8930F05A94C42BCD7D5100A4FC7_8_1734423903.png)\n\nКакое из следующих неравенств верно?\n\n1) $xy>0$\n2) $x^2y<0$\n3) $x+y>0$\n4) $x-y<0$","answer_text":"4","answer":"4","solution_text":"По рисунку точка $x$ лежит **левее** нуля, а точка $y$ — **правее** нуля, причём $x$ удалена от нуля дальше, чем $y$. Значит:\n\n$$x<0,\\qquad y>0,\\qquad |x|>|y|.$$\n\nПроверим каждое неравенство.\n\n1) $xy>0$. Произведение отрицательного числа на положительное отрицательно, то есть $xy<0$. Неверно.\n\n2) $x^2y<0$. Так как $x\\ne 0$, то $x^2>0$; произведение положительных чисел $x^2$ и $y$ положительно. Неверно.\n\n3) $x+y>0$. Знак суммы чисел разных знаков совпадает со знаком слагаемого с большим модулем, а $|x|>|y|$, поэтому $x+y<0$. Неверно.\n\n4) $x-y<0$. Здесь $x<0$ и $-y<0$, а сумма двух отрицательных чисел отрицательна, значит $x-y<0$. Верно.\n\n**Проверка подстановкой.** Возьмём числа, отвечающие рисунку: $x=-3$, $y=1$. Тогда $xy=-3<0$; $x^2y=9>0$; $x+y=-2<0$; $x-y=-4<0$. Верно только четвёртое неравенство.\n\n**Ответ:** 4.","explanation":null,"score":1,"part":"1","metadata":{"answer_confidence":"high","answer_input_type":"choice_one","classify_confidence":"high","fipi_guid":"AD9FB8930F05A94C42BCD7D5100A4FC7","fipi_short":"6985CA","qkind":"ILI_STD_SELECTONE","source":"fipi_oge_bank","tags":["координатная прямая","знаки чисел","сравнение чисел","числовые неравенства","модуль числа"]},"created_at":"2026-07-27T07:58:27.705002Z","updated_at":"2026-08-31T15:37:35.489242Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","name":"Неравенства","description":null},"tags":[{"id":"0148c122-19f0-472b-b8c1-162241eb57d1","code":"tag-2","name":"Модуль числа"},{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33a6ace9-99be-4b5e-9b2a-7716d2d006e8","code":"tag-980","name":"сравнение чисел"},{"id":"536cfc98-1d12-4e75-a605-bf4e8216ce2e","code":"tag-780","name":"числовые неравенства"},{"id":"be65932e-76f1-4635-8f6a-d290fdfdf2f0","code":"tag-417","name":"координатная прямая"},{"id":"da06b24c-f6be-4d02-bbb8-4bfc6dd93f9e","code":"tag-382","name":"знаки чисел"},{"id":"da7c43d6-c792-49b4-ae18-fad888360023","code":"tag-966","name":"модуль числа"}]}},{"problem_id":"a87fe749-d60a-4e7e-b556-4258c29d31e8","collection_id":"82875418-ca33-4805-bb13-87a9b13be4ed","display_order":2,"metadata":{},"created_at":"2026-07-27T07:58:53.818369Z","updated_at":"2026-07-27T07:58:53.818369Z","problem":{"id":"a87fe749-d60a-4e7e-b556-4258c29d31e8","short_id":20811,"task_number":7,"topic_id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","exam_id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","difficulty":2,"content_text":"На координатной прямой отмечены числа ![x](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/616F27CC18FFB8F84F153BC622EDC089/xs3qstsrc616F27CC18FFB8F84F153BC622EDC089_6_1734423564.png) и ![y](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/616F27CC18FFB8F84F153BC622EDC089/xs3qstsrc616F27CC18FFB8F84F153BC622EDC089_7_1734423564.png).\n\n![Координатная прямая: точка x левее нуля, точка y правее нуля, расстояние от нуля до x примерно втрое больше расстояния от нуля до y](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/616F27CC18FFB8F84F153BC622EDC089/xs3qstsrc616F27CC18FFB8F84F153BC622EDC089_8_1734423564.png)\n\nКакое из следующих неравенств верно?\n\n1) $xy>0$\n\n2) ${x}^{2}y<0$\n\n3) $x+y<0$\n\n4) $x-y\\geqslant 0$","answer_text":"3","answer":"3","solution_text":"**Читаем чертёж.**\n\nТочка \$x\$ отмечена левее нуля, поэтому \$x<0\$. Точка \$y\$ отмечена правее нуля, поэтому \$y>0\$. Кроме того, от нуля до \$x\$ заметно дальше (примерно втрое), чем от нуля до \$y\$, поэтому \$|x|>|y|\$.\n\n**Проверяем каждое неравенство.**\n\n1) \$xy>0\$ — неверно: произведение отрицательного \$x\$ и положительного \$y\$ отрицательно, то есть \$xy<0\$.\n\n2) \$x^{2}y<0\$ — неверно: \$x^{2}>0\$ и \$y>0\$, значит \$x^{2}y>0\$.\n\n3) \$x+y<0\$ — верно: \$x+y=-|x|+|y|=-\\left(|x|-|y|\\right)\$, а \$|x|-|y|>0\$, поэтому сумма отрицательна.\n\n4) \$x-y\\geqslant 0\$ — неверно: \$x<0\$ и \$-y<0\$, значит \$x-y<0\$.\n\n**Проверка подстановкой.**\n\nВозьмём числа, расположенные так же, как на чертеже: \$x=-3\$, \$y=1\$. Тогда \$xy=-3<0\$; \$x^{2}y=9>0\$; \$x+y=-2<0\$; \$x-y=-4<0\$. Верным оказывается только третье неравенство.\n\n**Ответ:** 3","explanation":null,"score":1,"part":"1","metadata":{"answer_confidence":"high","answer_input_type":"choice_one","classify_confidence":"high","fipi_guid":"616F27CC18FFB8F84F153BC622EDC089","fipi_short":"F08D22","qkind":"ILI_STD_SELECTONE","source":"fipi_oge_bank","tags":["координатная прямая","числовые неравенства","знаки чисел","модуль числа","сравнение чисел"]},"created_at":"2026-07-27T07:58:27.705002Z","updated_at":"2026-08-31T15:37:35.489242Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","name":"Неравенства","description":null},"tags":[{"id":"0148c122-19f0-472b-b8c1-162241eb57d1","code":"tag-2","name":"Модуль числа"},{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33a6ace9-99be-4b5e-9b2a-7716d2d006e8","code":"tag-980","name":"сравнение чисел"},{"id":"536cfc98-1d12-4e75-a605-bf4e8216ce2e","code":"tag-780","name":"числовые неравенства"},{"id":"be65932e-76f1-4635-8f6a-d290fdfdf2f0","code":"tag-417","name":"координатная прямая"},{"id":"da06b24c-f6be-4d02-bbb8-4bfc6dd93f9e","code":"tag-382","name":"знаки чисел"},{"id":"da7c43d6-c792-49b4-ae18-fad888360023","code":"tag-966","name":"модуль числа"}]}},{"problem_id":"c930359d-b35a-4ab6-bbab-0fc45caa6053","collection_id":"82875418-ca33-4805-bb13-87a9b13be4ed","display_order":3,"metadata":{},"created_at":"2026-07-27T07:58:53.818369Z","updated_at":"2026-07-27T07:58:53.818369Z","problem":{"id":"c930359d-b35a-4ab6-bbab-0fc45caa6053","short_id":20814,"task_number":7,"topic_id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","exam_id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","difficulty":2,"content_text":"На координатной прямой отмечены числа $x$ и $y$.\n\n![Координатная прямая: точка y слева от нуля и близко к нему, точка x справа от нуля и заметно дальше](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/F551E146497595FC47026621824F850A/xs3qstsrcF551E146497595FC47026621824F850A_6_1734423722.png)\n\nКакое из следующих неравенств верно?\n\n1) $x+y<0$\n2) $xy<0$\n3) $y-x>0$\n4) $x^2y>0$","answer_text":"2","answer":"2","solution_text":"По рисунку точка $y$ лежит **левее** нуля и близко к нему, а точка $x$ — **правее** нуля и заметно дальше от него. Значит:\n\n$$y<0,\\qquad x>0,\\qquad |y|<|x|.$$\n\nПроверим каждое неравенство.\n\n1) $x+y<0$. Знак суммы чисел разных знаков совпадает со знаком слагаемого с большим модулем, а $|x|>|y|$, поэтому $x+y>0$. Неверно.\n\n2) $xy<0$. Произведение положительного числа $x$ на отрицательное $y$ отрицательно. Верно.\n\n3) $y-x>0$. Здесь $y<0$ и $-x<0$, а сумма двух отрицательных чисел отрицательна, значит $y-x<0$. Неверно.\n\n4) $x^2y>0$. Так как $x\\ne 0$, то $x^2>0$; произведение положительного $x^2$ на отрицательное $y$ отрицательно. Неверно.\n\n**Проверка подстановкой.** Возьмём числа, отвечающие рисунку: $x=3$, $y=-1$. Тогда $x+y=2>0$; $xy=-3<0$; $y-x=-4<0$; $x^2y=-9<0$. Верно только второе неравенство.\n\n**Ответ:** 2.","explanation":null,"score":1,"part":"1","metadata":{"answer_confidence":"high","answer_input_type":"choice_one","classify_confidence":"high","fipi_guid":"F551E146497595FC47026621824F850A","fipi_short":"13FF81","qkind":"ILI_STD_SELECTONE","source":"fipi_oge_bank","tags":["координатная прямая","знаки чисел","сравнение чисел","числовые неравенства","модуль числа"]},"created_at":"2026-07-27T07:58:27.705002Z","updated_at":"2026-08-31T15:37:35.489242Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","name":"Неравенства","description":null},"tags":[{"id":"0148c122-19f0-472b-b8c1-162241eb57d1","code":"tag-2","name":"Модуль числа"},{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33a6ace9-99be-4b5e-9b2a-7716d2d006e8","code":"tag-980","name":"сравнение чисел"},{"id":"536cfc98-1d12-4e75-a605-bf4e8216ce2e","code":"tag-780","name":"числовые неравенства"},{"id":"be65932e-76f1-4635-8f6a-d290fdfdf2f0","code":"tag-417","name":"координатная прямая"},{"id":"da06b24c-f6be-4d02-bbb8-4bfc6dd93f9e","code":"tag-382","name":"знаки чисел"},{"id":"da7c43d6-c792-49b4-ae18-fad888360023","code":"tag-966","name":"модуль числа"}]}},{"problem_id":"ee80e95a-4c0f-48a3-b871-c46964006562","collection_id":"82875418-ca33-4805-bb13-87a9b13be4ed","display_order":4,"metadata":{},"created_at":"2026-07-27T07:58:53.818369Z","updated_at":"2026-07-27T07:58:53.818369Z","problem":{"id":"ee80e95a-4c0f-48a3-b871-c46964006562","short_id":20813,"task_number":7,"topic_id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","exam_id":"b0db9506-ddcf-4200-8815-7a709ea0dc4e","difficulty":2,"content_text":"На координатной прямой отмечены числа ![a](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/BF3B5CB29CEA9F1C453FE75D3DE6ADED/xs3qstsrcBF3B5CB29CEA9F1C453FE75D3DE6ADED_6_1734423580.png) и ![b](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/BF3B5CB29CEA9F1C453FE75D3DE6ADED/xs3qstsrcBF3B5CB29CEA9F1C453FE75D3DE6ADED_7_1734423580.png).\n\n![Координатная прямая: точка a слева от нуля и близко к нему, точка b справа от нуля и заметно дальше](https://db.sdamex.com/storage/v1/object/public/math_prof_imgs/oge/BF3B5CB29CEA9F1C453FE75D3DE6ADED/xs3qstsrcBF3B5CB29CEA9F1C453FE75D3DE6ADED_8_1734423580.png)\n\nКакое из следующих неравенств верно?\n\n1) $a+b<0$\n2) $b-a<0$\n3) $ab<0$\n4) $ab^2>0$","answer_text":"3","answer":"3","solution_text":"По рисунку точка $a$ лежит **левее** нуля и близко к нему, а точка $b$ — **правее** нуля и заметно дальше от него. Значит:\n\n$$a<0,\\qquad b>0,\\qquad |a|<|b|.$$\n\nПроверим каждое неравенство.\n\n1) $a+b<0$. Знак суммы чисел разных знаков совпадает со знаком слагаемого с большим модулем, а $|b|>|a|$, поэтому $a+b>0$. Неверно.\n\n2) $b-a<0$. Здесь $b>0$ и $-a>0$, а сумма двух положительных чисел положительна, значит $b-a>0$. Неверно.\n\n3) $ab<0$. Произведение отрицательного числа на положительное отрицательно. Верно.\n\n4) $ab^2>0$. Так как $b\\ne 0$, то $b^2>0$; произведение отрицательного $a$ на положительное $b^2$ отрицательно, то есть $ab^2<0$. Неверно.\n\n**Проверка подстановкой.** Возьмём числа, отвечающие рисунку: $a=-1$, $b=3$. Тогда $a+b=2>0$; $b-a=4>0$; $ab=-3<0$; $ab^2=-9<0$. Верно только третье неравенство.\n\n**Ответ:** 3.","explanation":null,"score":1,"part":"1","metadata":{"answer_confidence":"high","answer_input_type":"choice_one","classify_confidence":"high","fipi_guid":"BF3B5CB29CEA9F1C453FE75D3DE6ADED","fipi_short":"819E97","qkind":"ILI_STD_SELECTONE","source":"fipi_oge_bank","tags":["координатная прямая","знаки чисел","сравнение чисел","числовые неравенства","модуль числа"]},"created_at":"2026-07-27T07:58:27.705002Z","updated_at":"2026-08-31T15:37:35.489242Z","is_favorite":false,"is_solved":false,"is_reported":false,"topic":{"id":"45db12ec-15f0-4bb6-85e9-c2fe3e8b440b","name":"Неравенства","description":null},"tags":[{"id":"0148c122-19f0-472b-b8c1-162241eb57d1","code":"tag-2","name":"Модуль числа"},{"id":"31d4e4b7-a2c1-4cbd-85ff-1aa0dec4e314","code":"tag-45","name":"Сравнение чисел"},{"id":"33a6ace9-99be-4b5e-9b2a-7716d2d006e8","code":"tag-980","name":"сравнение чисел"},{"id":"536cfc98-1d12-4e75-a605-bf4e8216ce2e","code":"tag-780","name":"числовые неравенства"},{"id":"be65932e-76f1-4635-8f6a-d290fdfdf2f0","code":"tag-417","name":"координатная прямая"},{"id":"da06b24c-f6be-4d02-bbb8-4bfc6dd93f9e","code":"tag-382","name":"знаки чисел"},{"id":"da7c43d6-c792-49b4-ae18-fad888360023","code":"tag-966","name":"модуль числа"}]}}],"total":5},"serverExams":[{"id":"a217734d-d4f5-4d66-ba1a-5248bb381f09","code":"dvi-math","name":"ДВИ МГУ (математика)","subject_id":"d1be5154-7be0-4c8c-92d3-c486ea0a1a57","subject_name":"Математика","subject_code":"math","program_id":"1aed6121-06cc-4a5a-87f5-77de484ebb36","program_code":"dvi","exam_type":"ДВИ","description":"Дополнительное вступительное испытание МГУ им. 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